The local Bernoulli limit conjecture for random tiles of $\mathbb{Z}^d$

Fix an integer $k$, and pick $x \in [n]^d$ uniformly at random. Let $\mathcal{S}$ be a random tile contained in $[n]$, and let $B_k$ be the $k$-ball around $x$. Local Bernoulli limit conjecture. The local limit of $\left.\mathcal{S}\right|_{B_k}$ as $n$ tends to infinity is a product measure of Bernoulli $\frac{1}{3}$ random variables. Here, “local limit” is understood in the sense of the definition cited by the source. This conjecture predicts asymptotic local independence and density $\frac{1}{3}$ for a uniformly sampled random tile; the source presents it among its concluding conjectures and gives no resolution.